Abstract
The problem on leaky waves in an anisotropic inhomogeneous dielectric waveguide is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Yu. G. Smirnov, ‘‘Application of the operator pencil method in the eigenvalue problem for partially,’’ Sov. Phys. Dokl. 35, 430 (1990).
Yu. G. Smirnov, ‘‘The method of operator pencils in the boundary transmission problems for elliptic system of equations,’’ Differ. Equat. 27, 140–147 (1991).
Yu. G. Smirnov, Mathematical Methods for Electromagnetic Problems (Penz. Gos. Univ., Penza, 2009) [in Russian].
Y. Shestopalov and Y. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Spectrum,’’ Applic. Anal. 93, 408–427 (2014).
Y. Shestopalov and Y. Smirnov, ‘‘Eigenwaves in waveguides with dielectric inclusions: Completeness,’’ Applic. Anal. 93, 1824–1845 (2014).
M. V. Keldysh, ‘‘On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators,’’ Dokl. Akad. Nauk SSSR 77, 11–14 (1951).
A. L. Delitsyn, ‘‘An approach to the completeness of normal waves in a waveguide with magnitodielectric filling,’’ Differ. Equat. 36, 695–700 (2000).
P. E. Krasnushkin and E. I. Moiseev, ‘‘On the excitation of oscillations in layered radiowaveguide,’’ Sov. Phys. Dokl. 27, 458 (1982).
A. S. Zilbergleit and Yu. I. Kopilevich, Spectral Theory of Guided Waves (Inst. Phys. Publ., London, 1966).
V. V. Lozhechko and Yu. V. Shestopalov, ‘‘Problems of the excitation of open cylindrical resonators with an irregular boundary,’’ Comput. Math. Math. Phys. 35, 53–61 (1995).
G. I. Veselov and S. B. Raevskii, Metal-Dielectric Waveguides Formed by Layers (Radio Svyaz, Moscow, 1988) [in Russian].
L. Levin, Theory of Waveguides (Newnes-Butterworths, London, 1975).
Yu. G. Smirnov and E. Smolkin, ‘‘Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide,’’ Differ. Equat. 53, 1168–1179 (2017).
Yu. G. Smirnov, E. Smolkin, and M. O. Snegur, ‘‘Analysis of the spectrum of azimuthally symmetric waves of an open inhomogeneous anisotropic waveguide with longitudinal magnetization,’’ Comput. Math. Math. Phys. 58, 1887–1901 (2018).
Yu. G. Smirnov and E. Smolkin, ‘‘Operator function method in the problem of normal waves in an inhomogeneous waveguide,’’ Differ. Equat. 54, 1262–1273 (2018).
Yu. G. Smirnov and E. Smolkin, ‘‘Investigation of the spectrum of the problem of normal waves in a closed regular inhomogeneous dielectric waveguide of arbitrary cross section,’’ Dokl. Math. 97, 86–89 (2017).
Yu. G. Smirnov and E. Smolkin, ‘‘Eigenwaves in a lossy metal-dielectric waveguide,’’ Applic. Anal. 4, 1–12 (2018).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965).
R. A. Adams, Sobolev Spaces (Academic, New York, 1975).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (Nauka, Moscow, 1965) [in Russian].
Funding
This work was supported by the Russian Science Foundation, project no. 20-11-20087.
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by E. E. Tyrtyshnikov)
Rights and permissions
About this article
Cite this article
Smolkin, E., Snegur, M. Mathematical Theory of Leaky Waves in an Anisotropic Waveguide. Lobachevskii J Math 43, 1285–1292 (2022). https://doi.org/10.1134/S1995080222080327
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222080327